Uniqueness of symmetric steady subsonic flows in infinitely long divergent nozzles

We prove that the spherically symmetric subsonic flows in an infinitely long straight divergent nozzle with arbitrary smooth cross-section are unique for the three-dimensional steady potential flow equation. The proof depends on an extreme principle for elliptic equations in an unbounded conical domain, under the assumption that the gradient of the solution is of order \({O\left(\frac{1}{|x|}\right)}\) as \({|x|\rightarrow\infty}\) . Similar result holds for steady subsonic Euler flows in two-dimensional infinitely long straight divergent nozzles.